Problem 11
Question
(a) Let \(\mathbb{H}:=\\{z \in \mathbb{C} ; \quad \operatorname{Im} z>0\\}\) be the upper half-plane. Show: \(z \in \mathrm{H} \Longleftrightarrow-1 / z \in \mathbb{H} .\) (b) Assume \(z, a \in \mathbb{C}\). Show: \(\quad|1-z \bar{a}|^{2}-|z-a|^{2}=\left(1-|z|^{2}\right)\left(1-|a|^{2}\right) .\) Deduce: If \(|a|<1\), then $$ |z|<1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|<1 \quad \text { and } \quad|z|=1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|=1 $$
Step-by-Step Solution
Verified Answer
(a) True, because it reflects the imaginary condition; (b) Proves the condition using the equation abs squared.
1Step 1: Understand the Upper Half-plane Condition
The upper half-plane \( \mathbb{H} \) consists of all complex numbers \( z \) such that the imaginary part \( \operatorname{Im}(z) > 0 \).
2Step 2: Evaluate \(-1/z\) for \(z \in \mathbb{H}\)
If \( z = x + yi \) with \( y > 0 \), calculate \(-1/z = -1/(x + yi)\). Simplify this to \(-\frac{x}{x^2+y^2} - i \frac{y}{x^2+y^2}\). The imaginary part is \(-\frac{y}{x^2+y^2}\), which is negative.
3Step 3: Reflect Imaginary Condition
Since the imaginary part of \(-1/z\) is negative, implying \(-1/z \in \mathbb{H} \), it must entirely lie in the upper half-plane \( \mathbb{H} \) for \(-1/z \) when \( z \in \mathbb{H} \). Hence, \(z \in \mathbb{H} \Leftrightarrow -1/z \in \mathbb{H}\).
4Step 4: Apply the Modulus Properties in Complex Numbers
For the modulus property, \(|1 - z\bar{a}|^2 = (1 - z\bar{a})(1 - \bar{z}a) = 1 - z\bar{a} - \bar{z}a + |z|^2|a|^2\) and \(|z-a|^2 = (z-a)(\bar{z} - \bar{a}) = |z|^2 - z\bar{a} - \bar{z}a + |a|^2\).
5Step 5: Show Equality of the Given Equation
Combine results: \(|1 - z\bar{a}|^2 - |z-a|^2 = (1 - z\bar{a} - \bar{z}a + |z|^2|a|^2) - (|z|^2 - z\bar{a} - \bar{z}a + |a|^2) = |z|^2|a|^2 + 1 - |z|^2 - |a|^2\). This validates the equation \((1-|z|^2)(1-|a|^2)\).
6Step 6: Deduce the Condition for \(|z|<1\) and \(|z|=1\)
If \(|a|<1\), use \(|1 - z\bar{a}|^2 - |z-a|^2=(1-|z|^2)(1-|a|^2)\) to show \((1-|z|^2)<(1-|a|^2)\). For \(|z|<1\), depending on the simplification \(\left|\frac{z-a}{\bar{a}z-1}\right|<1\) holds and similar setup shows \(|z|=1\) corresponds to \(\left|\frac{z-a}{\bar{a}z-1}\right|=1\).
Key Concepts
Upper Half-planeComplex ModulusImaginary PartComplex Numbers
Upper Half-plane
The upper half-plane, commonly denoted as \( \mathbb{H} \), is a fascinating concept in complex analysis. It consists of all complex numbers \( z = x + yi \) where the imaginary part \( \,\operatorname{Im}(z)\, \) is positive, which means \( y > 0 \). In simpler terms, if you plot these numbers on a standard complex plane, they occupy the region above the real axis.
The concept is essential when contemplating mappings and transformations in complex analysis. Given \( z \in \mathbb{H} \), a specific property arises: \(-1/z\) will also reside in the upper half-plane. This intriguing property results from the calculations of \(-1/z\) producing a negative imaginary part, upon reflection of \( z \) itself.
This unexpectedly symmetric interplay between \( z \) and \(-1/z\) offers a profound insight into many complex mathematical structures and functions, enabling further exploration of complex mappings and hyperbolic geometries.
The concept is essential when contemplating mappings and transformations in complex analysis. Given \( z \in \mathbb{H} \), a specific property arises: \(-1/z\) will also reside in the upper half-plane. This intriguing property results from the calculations of \(-1/z\) producing a negative imaginary part, upon reflection of \( z \) itself.
This unexpectedly symmetric interplay between \( z \) and \(-1/z\) offers a profound insight into many complex mathematical structures and functions, enabling further exploration of complex mappings and hyperbolic geometries.
Complex Modulus
The modulus of a complex number is like its length or distance from the origin in the complex plane. For any complex number \( z = x + yi \), the modulus \( |z| \) is calculated as \( \sqrt{x^2 + y^2} \). This makes it easy to determine the ‘size’ or ‘magnitude’ of \( z \).
Understanding and manipulating the modulus is crucial in many complex number operations. For example, calculating expressions such as \(|1 - z\bar{a}|^2\) and \(|z-a|^2\) involves using the modulus properties. Here, \( a \) is another complex number, and \( \bar{a} \) is its conjugate.
The modulus properties help illustrate relationships within complex numbers, especially when working with transformations in the unit circle and establishing conditions like \(|z| < 1\), which relates to the interior of the circle in the complex plane.
Understanding and manipulating the modulus is crucial in many complex number operations. For example, calculating expressions such as \(|1 - z\bar{a}|^2\) and \(|z-a|^2\) involves using the modulus properties. Here, \( a \) is another complex number, and \( \bar{a} \) is its conjugate.
The modulus properties help illustrate relationships within complex numbers, especially when working with transformations in the unit circle and establishing conditions like \(|z| < 1\), which relates to the interior of the circle in the complex plane.
Imaginary Part
The imaginary part of a complex number is a crucial component for representing numbers in the complex plane. If you express a complex number as \( z = x + yi \), \( y \) is the imaginary part. This part is essential as it helps determine the placement and behavior of complex numbers.
In many exercises, such as determining whether a number belongs to the upper half-plane, analyzing the imaginary part is fundamental. For example, if \( z \) retains a positive imaginary part, \( z \) resides in the upper half-plane. Transformations and reflections often hinge on the sign and magnitude of the imaginary component, offering insights into the symmetry and mapping properties of complex numbers.
Moreover, calculations like \(-1/z = -1/(x + yi)\) particularly require analyzing the imaginary term, effectively resolving its effects on transformations and extensions beyond basic algebraic manipulation.
In many exercises, such as determining whether a number belongs to the upper half-plane, analyzing the imaginary part is fundamental. For example, if \( z \) retains a positive imaginary part, \( z \) resides in the upper half-plane. Transformations and reflections often hinge on the sign and magnitude of the imaginary component, offering insights into the symmetry and mapping properties of complex numbers.
Moreover, calculations like \(-1/z = -1/(x + yi)\) particularly require analyzing the imaginary term, effectively resolving its effects on transformations and extensions beyond basic algebraic manipulation.
Complex Numbers
Complex numbers are numbers that have both real and imaginary parts, usually represented in the form \( z = x + yi \), where \( x \) and \( y \) are real numbers and \( i \) is the imaginary unit \( i^2 = -1 \). They allow for a richer set of operations and extensions beyond the real numbers.
In problems involving complex numbers, various operations are used, such as addition, subtraction, multiplication, and division, each defined in a way that combines real and imaginary components. For instance, dividing complex numbers often needs rationalization by multiplying by the conjugate, as seen with expressions like \( \frac{z-a}{\bar{a}z-1} \).
Complex numbers extend past mere arithmetic. They're used to explore mapping properties, like identifying units within the unit circle \(|z| = 1\) and examining conditions under which transformations maintain or alter their fundamental properties, as reflected in both the modulus and the behavior of their real and imaginary components.
In problems involving complex numbers, various operations are used, such as addition, subtraction, multiplication, and division, each defined in a way that combines real and imaginary components. For instance, dividing complex numbers often needs rationalization by multiplying by the conjugate, as seen with expressions like \( \frac{z-a}{\bar{a}z-1} \).
Complex numbers extend past mere arithmetic. They're used to explore mapping properties, like identifying units within the unit circle \(|z| = 1\) and examining conditions under which transformations maintain or alter their fundamental properties, as reflected in both the modulus and the behavior of their real and imaginary components.
Other exercises in this chapter
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